172 
THE MASS-LUMINOSITY RELATION 
knowledge is brought to bear on observation. But we have now reached 
the conclusion that the density is not incredible, and have some inclination 
to accept the straightforward calculation. Some difficulties remain— 
sufficiently impressive to deter us from accepting the high density as 
proved without further confirmation. 
I do not see how a star which has once got into this compressed con 
dition is ever going to get out of it. So far as we know, the close packing 
of matter is only possible so long as the temperature is great enough to 
ionise the material. When the star cools down and regains the normal 
density ordinarily associated with solids, it must expand and do work 
against gravity. The star will need energy in order to cool. Sirius comes on 
solidifying will have to expand its radius at least tenfold, which means 
that 90 per cent, of its lost gravitational energy Q must be replaced. We 
have seen (§§ 103, 104) that the heat energy including energy of ionisation 
is necessarily less than Q so that there is likely to be a deficit. We can 
scarcely credit the star with sufficient foresight to retain more than 90 per 
cent, in reserve for the difficulty awaiting it. It would seem that the 
star will be in an awkward predicament when its supply of sub-atomic 
energy ultimately fails. Imagine a body continually losing heat but with 
insufficient energy to grow cold ! 
It is a curious problem and one may make many fanciful suggestions 
as to what actually will happen. We here leave aside the difficulty as not 
necessarily fatal. 118 
118. The density of the companion of Sirius can be submitted to a 
crucial observational test, viz. the third Einstein effect or shift of spectral 
lines to the red. If the high density is right this effect will be very large 
since it is proportional to M/R which is 31 times as great for the star as 
for the sun. The predicted shift is equivalent to a Doppler displacement of 
20 km. per sec., and there is no fear of confusing it with miscellaneous 
sources of spectral shift (the K term) which can scarcely exceed 3 or 4 km. 
per second. In an isolated star there would be no means of separating 
the Einstein shift from a genuine Doppler displacement due to line-of- 
sight velocity; but for this star we know the line-of-sight velocity by 
observation of Sirius itself. The observation in fact consists in differential 
measures of the spectra of Sirius and its companion; the small difference 
of orbital motion between them is known and can be allowed for. 
This test has been carried out by W. S. Adams at the Mount Wilson 
Observatory*. Difficulty arises from the faintness of the object and its 
nearness to Sirius. The spectrum of the companion is overlaid by a scattered 
spectrum of Sirius. Scattering increases rapidly with diminishing wave 
length so that the long wave-length end of the spectrum is the purest. 
* Proc. Nat. Acad. Sci. 11, p. 382 (July, 1925); erratum, Observatory, 49, p. 88.